Theorem elpglem2 42455

Description: Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.)

Assertion

Ref	Expression
elpglem2	|- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )

Distinct variable group:   x, A

Proof of Theorem elpglem2

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- ( 1st ` A ) e. _V
2		fvex 6201	. . . . 5 |- ( 2nd ` A ) e. _V
3	1, 2	unex 6956	. . . 4 |- ( ( 1st ` A ) u. ( 2nd ` A ) ) e. _V
4	3	isseti 3209	. . 3 |- E. x x = ( ( 1st ` A ) u. ( 2nd ` A ) )
5		sseq1 3626	. . . . . 6 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( x C_ Pg <-> ( ( 1st ` A ) u. ( 2nd ` A ) ) C_ Pg ) )
6		unss 3787	. . . . . 6 |- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) <-> ( ( 1st ` A ) u. ( 2nd ` A ) ) C_ Pg )
7	5, 6	syl6bbr 278	. . . . 5 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( x C_ Pg <-> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) ) )
8	7	biimprd 238	. . . 4 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> x C_ Pg ) )
9		ssun1 3776	. . . . . . 7 |- ( 1st ` A ) C_ ( ( 1st ` A ) u. ( 2nd ` A ) )
10		id 22	. . . . . . 7 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> x = ( ( 1st ` A ) u. ( 2nd ` A ) ) )
11	9, 10	syl5sseqr 3654	. . . . . 6 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( 1st ` A ) C_ x )
12		vex 3203	. . . . . . 7 |- x e. _V
13	12	elpw2 4828	. . . . . 6 |- ( ( 1st ` A ) e. ~P x <-> ( 1st ` A ) C_ x )
14	11, 13	sylibr 224	. . . . 5 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( 1st ` A ) e. ~P x )
15		ssun2 3777	. . . . . . 7 |- ( 2nd ` A ) C_ ( ( 1st ` A ) u. ( 2nd ` A ) )
16	15, 10	syl5sseqr 3654	. . . . . 6 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( 2nd ` A ) C_ x )
17	12	elpw2 4828	. . . . . 6 |- ( ( 2nd ` A ) e. ~P x <-> ( 2nd ` A ) C_ x )
18	16, 17	sylibr 224	. . . . 5 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( 2nd ` A ) e. ~P x )
19	14, 18	jca 554	. . . 4 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) )
20	8, 19	jctird 567	. . 3 |- ( x = ( ( 1st ` A ) u. ( 2nd ` A ) ) -> ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
21	4, 20	eximii 1764	. 2 |- E. x ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )
22	21	19.37iv 1911	1 |- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   u. cun 3572   C_ wss 3574   ~Pcpw 4158   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  Pgcpg 42452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by:  elpg  42457